Question: Simplify; express your answer in exponential form. Assume $z\neq 0, t\neq 0$. $\dfrac{{(z^{2}t^{2})^{4}}}{{(z^{-2}t^{4})^{-2}}}$
Answer: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(z^{2}t^{2})^{4} = (z^{2})^{4}(t^{2})^{4}}$ On the left, we have ${z^{2}}$ to the exponent ${4}$ . Now ${2 \times 4 = 8}$ , so ${(z^{2})^{4} = z^{8}}$ Apply the ideas above to simplify the equation. $\dfrac{{(z^{2}t^{2})^{4}}}{{(z^{-2}t^{4})^{-2}}} = \dfrac{{z^{8}t^{8}}}{{z^{4}t^{-8}}}$ Break up the equation by variable and simplify. $\dfrac{{z^{8}t^{8}}}{{z^{4}t^{-8}}} = \dfrac{{z^{8}}}{{z^{4}}} \cdot \dfrac{{t^{8}}}{{t^{-8}}} = z^{{8} - {4}} \cdot t^{{8} - {(-8)}} = z^{4}t^{16}$